Euclid's Proof of the Infinitude of Primes

Euclid’s Proof of the Infinitude of Primes: Distorted, Clarified, Made Obsolete, and Confirmed in Modern Mathematics REINHARD SIEGMUND-SCHULTZE n this article1 I reflect on the recurrent theme of Michael Hardy and Catherine Woodgold (henceforth modernizing historical mathematical proofs, vocabulary, H&W) entitled ‘‘Prime Simplicity’’ appeared in this journal II and symbolism, and the extent to which this modern- in 2009. H&W discuss Euclid’s proof mainly from the ization serves to clarify, is able to preserve, or is bound to point of view of its distortion by modern authors who distort the original meaning. My example is also a quite often claim that it had the form of a reductio ad recurrent one: Euclid’s proof of the infinitude of primes absurdum.3 in book IX, theorem 20 of his Elements. Elementary My article, on the other hand, aims at a systematic pre- number theory is a very appropriate field for discussing sentation and logical analysis of Euclid’s proof (as it is such general historiographic questions on a nontechnical preserved in the critical editions by Euclid scholars) and, level.2 Indeed, a widely quoted and stimulating article by above all, at a more detailed discussion of its interpretation 1I dedicate this article to Walter Purkert (Bonn), the coordinator and very active force behind the excellent Felix Hausdorff edition soon to be completed, on the occasion of his 70th birthday in 2014. 2Mesˇ trovic´ (2012) contains many details on modern proofs of the infinitude of primes but is less interested in Euclid’s original theorem and its modern variants. Most of his material goes far beyond the simple fact of the infinitude of primes that is the focus of the present paper. 3In the following, the notions reductio ad absurdum, proof by contradiction, and indirect proof are used interchangeably for assuming the logical opposite of what one wants to prove and then reaching, by legitimate logical steps, a contradiction that requires the assumption of the original claim. Ó 2014 Springer Science+Business Media New York, Volume 36, Number 4, 2014 87 DOI 10.1007/s00283-014-9506-9 and rewriting by a few select modern mathematicians, that the Ancients ‘‘proved the irrationality of the square root foremost among them Dirichlet, G. H. Hardy, and Hilbert.4 of two’’ cannot be unqualifiedly maintained either. More As I will show, Euclid’s proof was constructive. For all recently, discussions were raised about the so-called the emphasis put on properties and axiomatic structure as ‘‘geometric algebra,’’ that is, the claim (for instance defen- displayed in Euclid’s Elements, constructions—finding, in ded by B. L. van der Waerden and meanwhile refuted by finitely many legitimate steps (i.e., those allowed by axioms historians) that some of Euclid’s geometric theorems have and theorems), a mathematical object with certain prop- to be interpreted as translations of historically preexisting erties—is the backbone of Greek mathematics.5 More algebraic problems and equations.6 specifically, I will argue that Euclid’s proof is ‘‘weakly Historians know, of course, that absolute historical constructive,’’ and it is partly this weak form of construc- ‘‘accuracy’’ cannot be attained in modern presentations and tiveness—the fact that Euclid does not provide an effective that they have to refer to modern vocabulary to make the method (formula) to calculate new primes from given very process of creation understandable.7 In discussing ones—which provided modern mathematicians with the Euclid’s Elements, the problem starts with the question as to wrong impression that Euclid’s proof was indirect. what extent it is legitimate to replace the purely verbal I agree with H&W that the insinuation of an indirect proof original formulation throughout the book by some other, in Euclid IX, 20 became the ‘‘prevailing doctrine’’ when modernized one that uses symbols. In the special case of modern mathematicians wanted to relate in a positive way to the Pythagorean-Euclidean number theory, Euclid’s dress- Euclid’s geometry. However, modern presentations of ing up of the theorems in geometric clothing, without Euclid’s proof, such as the ones by Hardy in 1938 and 1940, genuinely geometric content, adds to the problem. are not genuine indirect proofs of the infinitude of primes. In the case of Euclid’s Elements, one is best served They are quite close in spirit and content to Euclid’s original relying on Bernard Vitrac’s careful French edition (1990– proof. 2001) with very detailed commentary,8 which is still based (as is the most famous English edition by Thomas Heath What Does ‘‘Projecting Modern Mathematics from 1908 and 1925) on the Greek text, edited by the into the Past’’ Mean, and How Much Historical Danish historian J. L. Heiberg in 1883–1885.9 Vitrac says in Accuracy Can or Should One Reach in Modern his edition that—because of a scarcity of original sources— Texts? each modern specialist of Greek number theory provides a Mathematicians and historians projecting modern mathe- different interpretation (Vitrac 1994: 288). This should warn matics into the past can miss essential points of the us, the nonspecialists, to restrict our discussion basically to intentions and results of the historical creators of mathe- the modern reception of what the specialists agree is a matics. To mention a few: the modern notion of functional reliable text of Euclid’s theorem IX, 20 and other theorems dependence and its graphic representation originated related to it. (basically) in the Scientific Revolution of the 17th century, Vitrac’s and Heath’s commentaries (and others) do not connected to the need to extend the realm of mathematical contribute many answers to my questions: to what extent is objects to some ‘‘mechanical’’ curves (higher algebraic Euclid’s theorem IX, 20 constructive or its proof indirect, curves treated by Descartes) and to formulate the laws of what kind of infinity does it claim, and how has the physics (Newton). Projecting that modern notion uncriti- theorem been interpreted by modern mathematicians. cally into the past does not, however, help us much to Nevertheless I will try to use their and other Euclid scholars’ understand Apollonius’s sophisticated theory of conics. expertise to provide nuancing counterbalance against some Even Euclid’s Elements, which, because of their axiomatic formulations that necessarily have to be somewhat simpli- structure, have a more familiar appeal to modern mathe- fying so as to be short and understandable. maticians than Apollonius’s works, cause heated My first question is admittedly speculative and cannot be methodological-historical debates in their modern inter- answered in this article (although an answer would shed pretations. For example, the deep theory of geometric light on our problems): Why did Euclid prove the ‘‘infini- proportions in book V of the Elements caused R. Lipschitz, tude of primes’’ in the first place? in his discussion with R. Dedekind in the 1870s, to claim its The proof comes in book IX of the Elements, as prop- logical equivalence with the modern definition of real osition 20, almost at the end of the ‘‘number-theoretic’’ numbers. The latter, being the inventor of the Dedekind (‘‘Pythagorean’’) books VII through IX, whereas the deci- cut, of course denied that claim, and most historians agree sive instruments and propositions, particularly the with him (Nikolic´ 1974). One consequence is that the claim Euclidean algorithm and propositions close (though not 4The focus of the article is not, however, on the reception of Euclid’s work during the more than 2000 years since its creation, which would require special linguistic (including Arabic!), philologic, and philosophical qualifications. Although H&W refer, mostly critically and summarily, to an impressive number of 147 publications, I will quote many fewer modern mathematicians. However, I will refer to a few additional ones (Hilbert, Weil). In addition I will use historical literature, above all B. Vitrac’s edition of the Elements. 5More on that later, in connection with an analysis of Euclid’s proof. 6Concerns about this notion were already articulated in the late 1960s by A. Szabo´ , indicated in Knorr (1975), and formulated polemically by Unguru (1975). 7We will speak frequently, for instance, about ‘‘sets of primes’’ below, without, however, assuming modern set-theoretical operations and notions coming with it. 8Of course the French language might constitute a barrier to the reception of Vitrac’s edition in our modern, largely English-speaking world. We will in the following refer to volume 2 of this edition, containing books V–IX of the Elements, as Vitrac (1994). 9Cf. June Barrow-Green (2006) for an overview of earlier English editions of the Elements. 88 THE MATHEMATICAL INTELLIGENCER fully equivalent) to the fundamental theorem of arithmetic, substantially by contradiction. Various forms of MIP are presented in the beginning and in the middle of book (Modern Infinitude Proofs) sometimes are substantially VII. The algorithm and the propositions are frequently used indirect proofs (as in the cases of Dirichlet [1863] and Hardy in Euclid’s number-theoretic books, but his theorem on the [1908] to be discussed later) and sometimes claim to be ‘‘infinitude’’ of primes is not used at all. The remaining indirect although they follow closely the Euclidean original propositions of book IX, 21 through 36, are mainly about (as in the case of another proof by Hardy of 1938 and the properties of odd and even numbers.10 Thus theorem 1940). IX, 20 appears isolated.

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